Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

About the critical condition of the spin S=1 Ising model on the anisotropic square and SC lattice

Ilija M. Savić, L. J. Maurer

Published Dec 1, 1987• 0 citations
Publisher

Abstract

The authors extend to the S=1 Ising model a simple procedure which was proposed recently for the q-state Potts model. The approximate critical conditions are obtained for the ferromagnetic model on the square and simple cubic lattice.
DOI
10.1088/0305-4470/20/17/049

Related papers

Anisotropic cubic lattice Potts ferromagnet: renormalisation group treatment

L. R. da Silva, Constantino Tsallis, Georges Schwachheim

Within a real space renormalisation group framework, the authors discuss the criticality of the full anisotropic (arbitrary J x , J y and J z )q-state Potts ferromagnet in the simple cubic lattice. Several previously known exact results for the d=1 and d=2 particular cases are recovered. Furthermore they obtain: (i) the q-dependence of the d=3 correlation length critical exponent nu 3 (in particular, if q to 0, nu 3 (q) approximately nu 3 (0)+ nu 3 '(0)q where the present approximate values are nu 3 (0) approximately=1.105 and nu 3 '(0) approximately=-0.66); (ii) the q-dependence of the d=2 to or from d=3 crossover critical exponent phi 23 (in particular, phi 23 varies as 1/ square root q if q to 0); (iii) through a convenient numerical extrapolation, a quite accurate proposal for the critical temperatures corresponding to arbitrary ratios, J y /J x and J z /J x and values of q.

Journal of Physics A Mathematical and General•Nov 11, 1984•9 citations

The Critical Lines of the Yang—Lee Edge Singularities of the Anisotropic Ising Ferromagnets on Square,Triangular and Honeycomb Lattices

WANGXian-Zhi

Using the numerical results and some results from the renormalization group theory, we extend our previous approach of determining the Yang-Lee edge singularities of Ising ferromagnets on square, triangular and honeycomb lattices (Phys. Rev. Lett. 78 (1997) 413; Phys. Rev. E56 (1998) 2793; E57 (1998) 5013) and obtain accurate closed-form approximations of the critical lines of anisotropic Ising ferromagnets on these lattices.

理论物理通讯：英文版•Jan 1, 2001•0 citations

The Critical Lines of the Yang–Lee Edge Singularities of the Anisotropic Ising Ferromagnets on Square, Triangular and Honeycomb Lattices

Xianzhi Wang

Using the numerical results and some results from the renormalization group theory, we extend our previous approach of determining the Yang–Lee edge singularities of Ising ferromagnets on square, triangular and honeycomb lattices (Phys. Rev. Lett. 78 (1997) 413; Phys. Rev. E56 (1998) 2793; E57 (1998) 5013) and obtain accurate closed-form approximations of the critical lines of anisotropic Ising ferromagnets on these lattices.

Communications in Theoretical Physics•Mar 15, 2001•2 citations

Series studies of the Potts model. I. The simple cubic Ising model

A J Guttmann, I G Enting

The finite-lattice method of series expansion is generalized to the q-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2) case the authors have extended the low-temperature series for the partition functions, magnetization and zero-field susceptibility to u 26 from u 20 . The high-temperature series for the zero-field partition function is extended from nu 18 to nu 22 . Subsequent analysis gives critical exponents in agreement with those from field theory.

arXiv:hep-lat/9212032•Journal of Physics A Mathematical and General•Feb 21, 1993•49 citations

Equations of state of the Potts ferromagnet in an anisotropic square lattice: Renormalization-group approach

Anna Chame, Constantino Tsallis, Uriel M. S. Costa

We extend to the anisotropic case (not necessarily equal coupling constants along the various crystalline directions) the real-space renormalization-group procedure recently introduced by Caride and Tsallis, which directly yields the order parameter for arbitrary temperatures. The method does not demand the calculation of the thermodynamic energy; it is as simple as a mean-field calculation although it provides nontrivial results, which can be systematically improved. We apply the procedure to the q-state Potts ferromagnet in an anisotropic square lattice, whose exact equation of state is still unknown. The results compare satisfactorily with available exact results.

Physical review. B, Condensed matter•May 1, 1988•5 citations

Anisotropic simple-cubic Ising lattice: extended phenomenological renormalization-group treatment

M. A. Yurishchev

Using transfer-matrix extended phenomenological renormalization-group methods [M.A.Yurishchev, Nucl. Phys. B (Proc. Suppl.) 83-84, 727 (2000); hep-lat/9908019; J. Exp. Theor. Phys. 91, 332 (2000); cond-mat/0108002] the improved estimates for the critical temperature of spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths ${\vec J}=(J',J',J)$ are obtained. Universality of both fundamental critical exponents $y_t$ and $y_h$ is confirmed. We show also that the critical finite-size scaling amplitude ratios $A_{χ^{(4)}}A_κ/A_χ^2$, $A_{κ^{''}}/A_χ$, and $A_{κ^{(4)}}/A_{χ^{(4)}}$ are independent of the lattice anisotropy parameter $Δ=J'/J$.

arXiv:cond-mat/0312555•Dec 21, 2003•0 citations
Within a real space renormalisation group framework, the authors discuss the criticality of the full anisotropic (arbitrary J x , J y and J z )q-state Potts ferromagnet in the simple cubic lattice. Several previously known exact results for the d=1 and d=2 particular cases are recovered. Furthermore they obtain: (i) the q-dependence of the d=3 correlation length critical exponent nu 3 (in particular, if q to 0, nu 3 (q) approximately nu 3 (0)+ nu 3 '(0)q where the present approximate values are nu 3 (0) approximately=1.105 and nu 3 '(0) approximately=-0.66); (ii) the q-dependence of the d=2 to or from d=3 crossover critical exponent phi 23 (in particular, phi 23 varies as 1/ square root q if q to 0); (iii) through a convenient numerical extrapolation, a quite accurate proposal for the critical temperatures corresponding to arbitrary ratios, J y /J x and J z /J x and values of q.
The finite-lattice method of series expansion is generalized to the q-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2) case the authors have extended the low-temperature series for the partition functions, magnetization and zero-field susceptibility to u 26 from u 20 . The high-temperature series for the zero-field partition function is extended from nu 18 to nu 22 . Subsequent analysis gives critical exponents in agreement with those from field theory.
We extend to the anisotropic case (not necessarily equal coupling constants along the various crystalline directions) the real-space renormalization-group procedure recently introduced by Caride and Tsallis, which directly yields the order parameter for arbitrary temperatures. The method does not demand the calculation of the thermodynamic energy; it is as simple as a mean-field calculation although it provides nontrivial results, which can be systematically improved. We apply the procedure to the q-state Potts ferromagnet in an anisotropic square lattice, whose exact equation of state is still unknown. The results compare satisfactorily with available exact results.
Using transfer-matrix extended phenomenological renormalization-group methods [M.A.Yurishchev, Nucl. Phys. B (Proc. Suppl.) 83-84, 727 (2000); hep-lat/9908019; J. Exp. Theor. Phys. 91, 332 (2000); cond-mat/0108002] the improved estimates for the critical temperature of spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths J⃗=(J′,J′,J){\vec J}=(J',J',J)J=(J′,J′,J) are obtained. Universality of both fundamental critical exponents yty_tyt​ and yhy_hyh​ is confirmed. We show also that the critical finite-size scaling amplitude ratios Aχ(4)Aκ/Aχ2A_{χ^{(4)}}A_κ/A_χ^2Aχ(4)​Aκ​/Aχ2​, Aκ′′/AχA_{κ^{''}}/A_χAκ′′​/Aχ​, and Aκ(4)/Aχ(4)A_{κ^{(4)}}/A_{χ^{(4)}}Aκ(4)​/Aχ(4)​ are independent of the lattice anisotropy parameter Δ=J′/JΔ=J'/JΔ=J′/J.